To go from Milan to Lucca you need to change in Florence.
There are Eurostar trains every hour Milan to Florence and
the trip takes about 2 hours. I don't know the schedule from
Florence to Lucca but it should take less than a hour.

From Lucca to Cinqueterre you have to go to the seaside
first (Viareggio) and then catch a train to La Spezia.

There's no difference if you do the tour this way or the
other.

Do you already know where you'll stay and eat in Lucca? I go
there pretty often if you need help.